Abstract
In this paper we study the Schrödinger-Poisson system
where the potential \(V(x)\) and the weighted functions \(a(x), b(x)\) are positive and bounded in \(\mathbb {R}^{3}\), \(K(x)\in L^{2}(\mathbb {R}^{3})\cup L^{\infty}(\mathbb {R}^{3})\) and \(K(x)\ge0\) in \(\mathbb {R}^{3}\). We prove the existence of a positive solution \((u,\phi)\in W^{1,2}(\mathbb {R}^{3})\times\mathcal{D}^{1,2}(\mathbb {R}^{3})\) for \(4< q< m<2^{*}=6\) and \(\lambda\in \mathbb {R}\).
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1 Introduction and main results
In this paper, we study the existence of positive solutions for the Schrödinger-Poisson system
where \(V(x)\), \(a(x)\) and \(b(x)\) are positive and bounded in \(\mathbb {R}^{3}\), \(K(x)\in L^{2}(\mathbb {R}^{3})\cup L^{\infty}(\mathbb {R}^{3})\) and \(K(x)\ge0\) in \(\mathbb {R}^{3}\). We will prove the existence of a positive solution \((u,\phi)\in W^{1,2}(\mathbb {R}^{3})\times\mathcal{D}^{1,2}(\mathbb {R}^{3})\) for \(\lambda\in \mathbb {R}\) and \(4< q< m<2^{*}\), where \(2^{*}=6\) is the critical exponent for the Sobolev embedding in dimension 3. The assumption ‘\(4< q< m<6\)’ implies that the nonlinear term \(f(x,u)=a(x)|u|^{m-2}u+\lambda b(x)|u|^{q-2}u\) in (1.1) is superlinear, which is similar to those in [1].
Such a system, also known as the Schrödinger-Maxwell system, arises in many fields of physics. For example, the Schrödinger-Poisson system can describe the interaction of a charged particle with its own electrostatic field in quantum mechanics. The unknowns u and ϕ represent the wave functions associated with the particle and electric potential, and the functions V and K are, respectively, an external potential and nonnegative density charge. We refer to Benci and Fortunato [2] for more details on the physical aspects. This model can also appear in semiconductor theory to describe solitary waves [3].
In recent years, the Schrödinger-Poisson system
has been widely studied under various assumptions on V, K, and f via variational methods, and existence, nonexistence and multiplicity results have been obtained in many papers, see [4–10].
Very recently, Cerami and Vaira [11] considered problem (1.2) with \(K(x)\in L^{2}(\mathbb {R}^{3})\). They proved that (1.2) with \(V(x)=1\) and \(f(x,u)=a(x)|u|^{p-1}u\) (\(3< p<5\)) possesses a positive ground state solution by minimization on the Nehari manifold when \(a(x), K(x):\mathbb {R}^{3}\to \mathbb {R}\) are nonnegative functions such that
Similar results for \(V(x)=\lambda>0\) can be found in [4, 12, 13].
Liu et al. [14] also considered the existence of a solution for problem (1.2) with the potential \(V(x)\in C(\mathbb{R}^{3})\) satisfying \(\inf_{x\in\mathbb{R}^{3}}V(x)>-\infty\) and for every \(M>0\), \(\operatorname{meas}(\{x\in\mathbb{R}^{3} |V(x)\le M\})<\infty\). It is well known that this assumption guarantees that the embedding \(W^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{p}(\mathbb{R}^{3})\) is compact for each \(2\le p<6\). For problem (1.2), the function \(f(x,u)\) verifies \(uf(x,u)\ge4F(x,u)\) with \(F(x,u)=\int_{0}^{u}f(x,t)dt\). Similar assumptions can be found in [15–19].
However, to the best of our knowledge, there are few results on problem (1.1) when the potential \(V(x)\) and the weighted functions \(a(x)\), \(b(x)\) are bounded in \(\mathbb {R}^{N}\). In this paper, we are interested in the existence of a solution to problem (1.1) with \(V(x)\) satisfying
- (H1):
-
The function \(V(x)\in C(\mathbb{R}^{3})\) and \(0<\alpha _{0}:=\inf_{x\in\mathbb{R}^{3}}V(x)<\sup_{x\in\mathbb {R}^{3}}V(x)=:\alpha<\infty \);
- (H2):
-
\(\lim_{|x|\to\infty}V(x)=\alpha\).
Clearly, \(V(x)\) is not necessarily radial and coercive. For these assumptions, the embedding \(W^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{p}(\mathbb{R}^{3})\) is not compact. Furthermore, for problem (1.1), the function \(f(x,u)\) fails to satisfy the assumption \(uf(x,u)\ge4F(x,u)\). So the variational technique for problem (1.1) becomes more delicate. Arguing as in [20, 21], to preserve this compactness in some extent for our problem, we split a minimizing sequence \(\{u_{n}\}\) into two parts: \(u_{n}=u'_{n}+u''_{n}\) (\(n\in \mathbb{N}\)) such that \(u'_{n} \to u, u''_{n}\to0\) in \(L^{m}(\mathbb{R}^{3},a)\cap L^{q}(\mathbb{R}^{3},b)\). We will obtain a positive solution by using the Nehari manifold method.
In order to state our main results, we introduce some Sobolev spaces and norms. For \(p\ge1\), let \(L^{p}(\mathbb{R}^{3})\) be a usual Lebesgue space with the norm \(\|\cdot\|_{p}\). Denote
endowed with the norm
This norm is equivalent to the standard norm on \(W^{1,2}(\mathbb{R}^{3})\) under assumption (H1).
In general, let \(\|u\|_{p,\rho}=(\int_{\mathbb {R}^{3}}\rho|u|^{p}\,dx)^{1/p}\) with \(p\ge1\) and \(\rho=\rho(x)\ge0, \neq0\) a.e. in \(\mathbb {R}^{3}\). In particular, denote \(\|u\|_{p}=(\int_{\mathbb {R}^{3}}|u|^{p}\,dx)^{1/p}\) or \(\|u\|_{L^{p}(\Omega)}=(\int_{\Omega}|u|^{p}\,dx)^{1/p}\) with the domain \(\Omega\subset \mathbb {R}^{3}\).
Let \(\mathcal{D}^{1,2}(\mathbb {R}^{3})\) be the completion of \(C_{0}^{\infty}(\mathbb {R}^{3})\) with respect to the norm
The following Sobolev inequality [22] is well known. There is a constant \(S>0\) such that for every \(u\in \mathcal{D}^{1,2}(\mathbb{R}^{3})\),
Hence, inequality (1.4) holds in \(W^{1,2}(\mathbb{R}^{N})\) and E. Furthermore, there exists \(S_{p}>0\) such that for \(2\le p\le 6\),
It is well known that problem (1.1) can be reduced to a single equation with a nonlocal term, see [11, 23]. In fact, for every \(u\in E\), we define the linear functional \(\mathcal{L}_{u}\) by
If \(K(x)\in L^{\infty}(\mathbb {R}^{3})\), by the Hölder inequality and the Sobolev inequalities (1.4) and (1.5), we get
Similarly, if \(K\in L^{2}(\mathbb {R}^{3})\), we have
Hence, by the Lax-Milgram theorem, there exists a unique \(\phi_{u}\in \mathcal{D}^{1,2}(\mathbb {R}^{3})\) (see [11, 22]) such that
Moreover, \(\phi_{u}\) has the following integral expression:
and \(\phi_{u}(x)>0\) in \(\mathbb {R}^{3}\) if \(u\neq0\) and \(K(x)\ge0\). Therefore, we have from (1.10) that
If \(K\in L^{\infty}(\mathbb {R}^{3})\), it follows (1.6)-(1.9) that
and if \(K\in L^{2}(\mathbb {R}^{3})\), we have
where and in the sequel, \(d_{0}=\|K\|_{\infty}\). So, it follows from (1.12) and (1.13) that there exists a constant \(c_{1}>0\) such that
Furthermore, inserting \(\phi_{u}\) into the first equation in (1.1), we obtain
Let \(J(u):E\to \mathbb {R}\) be the energy functional associated to (1.15) defined by
where
The following assumptions will be used in this paper.
- (H3):
-
The parameters q, m and λ satisfy \(4< q< m<6\) and \(\lambda\in \mathbb {R}\).
- (H4):
-
The function \(K(x)\in L^{\infty}(\mathbb {R}^{3})\cup L^{2}(\mathbb {R}^{3})\) and \(K(x)\ge0\) in \(\mathbb {R}^{3}\).
- (H5):
-
The functions \(a(x),b(x)\in C(\mathbb {R}^{3})\) satisfy \(a_{0}\le a(x),b(x)\le a_{1}\) in \(\mathbb {R}^{3}\) for some constants \(a_{0},a_{1}>0\).
Under assumptions (H1)-(H5), it is easy to verify \(J\in C^{1}(E,\mathbb {R})\), and for any \(v \in E\), there holds
Hence, if \(u\in E\) is a critical point of J, that is, \(J'(u)v=0\) for \(\forall v\in E\), then the pair \((u,\phi_{u})\) is a solution of problem (1.1)(see [4, 23]). For the sake of simplicity in many cases, we just say that \(u\in E\), instead of \((u,\phi_{u})\in E\times\mathcal{D}^{1,2}(\mathbb {R}^{3})\), is a weak solution of problem (1.1).
Our main result in this paper is as follows.
Theorem 1.1
Assume that (H1)-(H5) hold. Then problem (1.1) admits at least a positive solution \(u\in E\).
Open problem
For \(K(x)\in L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{3})\), does equation (1.1) admit a positive solution \(u\in E\)? Denis and Carlo in [1] considered this problem with unbounded and vanishing potentials \(V(x)\). As far as we know, there is no result on the existence of positive solutions for (1.1) in the case \(K(x)\in L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{3})\) and \(V(x)\), \(a(x)\), \(b(x)\) satisfy (H1), (H2), (H3), (H5). Hence, this case should be also an interesting topic for future research.
This paper is organized as follows. In Section 2, we set up the variational framework and establish some lemmas, which will be used in the proof of Theorem 1.1. In Section 3, we prove Theorem 1.1.
2 Preliminaries
In this section, we are going to establish a series of lemmas to prove Theorem 1.1, in which we tacitly assume that the conditions in Theorem 1.1 are satisfied. We first set up the variational framework for problem (1.1).
Let \(J(u):E\rightarrow\mathbb{R} \) be the energy functional associated with problem (1.1) defined by (1.16), and its Gateaux derivative is given by (1.18).
Since the functional J is not bounded from below on E, a good candidate of an appropriate subset to study J is the so-called Nehari manifold for problem (1.1):
Notice that, if \(u\in\mathcal{N}\), then
Lemma 2.1
The Nehari manifold \(\mathcal{N}\neq\emptyset\).
Proof
Let \(u\in E\), \(u\not\equiv0\) in \(\mathbb{R}^{3}\). We consider the function
Since \(4< q< m\), it follows that \(h(t)>0\) for small \(t>0\) and \(h(t)\to-\infty\) as \(t\to\infty\). Then there exists \(t_{1}>0\) such that \(h(t_{1})=0\). Obviously, \(t_{1}u\not\equiv0\). Thus, we conclude that \(t_{1}u\in\mathcal{N}\) and \(\mathcal{N}\neq\emptyset\). □
Lemma 2.2
The functional J is coercive and bounded from below on \(\mathcal{N}\). Moreover,
Proof
Let \(u\in\mathcal{N}\). Then, from (1.4) and (2.1), it follows that
and
with some \(c_{1}>1\). If \(\|u\|_{E}\le1\), inequality (2.6) implies \(1\le2c_{1}\|u\|_{E}^{q-2}\). Hence,
If \(\|u\|_{E}\ge1\), inequality (2.6) gives \(1\le 2c_{1}\|u\|_{E}^{m-2}\) and
Hence, if \(u\in\mathcal{N}\), it follows from (2.7) and (2.8) that there exists \(c_{2}>0\) such that
Moreover, it follows from (2.2) that
This shows that the functional J is coercive and bounded from below on \(\mathcal{N}\) and \(d\ge c_{3}>0\). Then the proof of Lemma 2.2 is completed. □
Let \(\{u_{n}\}\) be a minimizing sequence for d in \(\mathcal{N}\), that is, \(J(u_{n})\to d\) as \(n\to\infty\) and
Furthermore, it follows from (2.2) that
This shows that \(\{u_{n}\}\) is bounded in E and, from (2.10), \(\{u_{n}\}\) is bounded in \(L^{m}(\mathbb{R}^{3},a)\) and \(L^{q}(\mathbb{R}^{3},b)\). Therefore, up to a subsequence, there exists \(u\in E\) such that as \(n\to\infty\),
with some \(M_{0}>0\). Since \(J(u_{n})=J(|u_{n}|)\), we assume \(u_{n}(x)\ge0\) a.e. in \(\mathbb{R}^{3}\) for every \(n\ge1\) and thus \(u(x)\ge0\) a.e. in \(\mathbb{R}^{3}\). By the weak lower semi-continuity of the norm, we get
By extracting a further subsequence, if necessary, we assume
Denote
By weak convergence, it is obvious that \(\theta\in[0, \beta]\). First, we have the following.
Lemma 2.3
There results \(\beta>0\).
Proof
Obviously, \(\beta\ge0\). If \(\beta=0\), we have \(\lim_{n\to\infty}\|u_{n}\|_{m,a}^{m}=0\). Let \(t\in(0,1)\) be such that \(q=2t+(1-t)m\). Then, by the Hölder inequality and (H5), we derive
with the constant \(b_{1}=a_{0}^{t-1}a_{1}\). Since \(\{u_{n}\}\) is bounded in E, so it is bounded in \(L^{p}(\mathbb{R}^{3})\) (\(2\le p\le6\)), and it follows from (2.16) that \(\lim_{n\to\infty}\|u_{n}\|_{q,b}^{q}=0\). Then (2.10) implies that \(u_{n}\to0\) in E and \(F(u_{n})\to0\) as \(n\to\infty\) and so \(d=0\). By Lemma 2.2, it is impossible. The proof of Lemma 2.3 is completed. □
Lemma 2.4
If \(\beta=\theta\), then \(u\in\mathcal{N}\) and \(J(u)=d\).
Proof
If \(\beta=\theta\), then \(\lim_{n\to\infty}\|u_{n}\|_{m,a}^{m}=\|u\|_{m,a}^{m}\). Since \(u_{n}\rightharpoonup u\) in \(L^{m}(\mathbb{R}^{3})\), it follows from the Brezis-Lieb lemma that \(u_{n}\to u\) in \(L^{m}(\mathbb{R}^{3},a)\). Similar to (2.16), we have
and \(u_{n}\to u\) in \(L^{q}(\mathbb{R}^{3},b)\). Hence, by the weak lower semi-continuity of the norm, we obtain
Furthermore, we have from (2.10) that
If the equality in (2.19) holds, then \(u\in\mathcal{N}\) and the lemma is proved.
We now assume that the equality in (2.19) fails to hold. Let
Clearly, \(h(t)>0\) for small \(t>0\) and \(h(1)<0\). Then there exists \(t\in(0,1)\) such that \(h(t)=0\), and then \(tu\in\mathcal{N}\) and
Here and in the sequel,
Relation (2.21) is a contradiction, and thus the proof of Lemma 2.4 is completed. □
We now turn to study the value of θ. First we introduce the Sobolev space
endowed with the norm
where α is the positive number defined in (H2). Consider the Schrödinger-Poisson system
The functional associated with problem (2.25) is
and the associated Nehari manifold is
Notice that \(u\in\mathcal{N}_{\alpha}\),
Finally, we define
Lemma 2.5
There results \(d< d_{\alpha}\), that is, \(\inf_{u\in \mathcal{N}}J(u)<\inf_{u\in \mathcal{N}_{\alpha}}J_{\alpha}(u)\).
Proof
Similar to the proofs of Lemmas 2.1 and 2.2, we can obtain that \(\mathcal{N}_{\alpha}\neq\emptyset\) and \(d_{\alpha}>0\). Arguing in Lemma 3.1.9 in [20], we see that there exists the nonnegative function \(u_{0}\in\mathcal{N}_{\alpha}\) such that \(J_{\alpha}(u_{0})=d_{\alpha}\). On the other hand, from (H1)-(H2), we infer
which implies that
Then we have
As the argument of Lemma 2.4, there exists \(t\in(0, 1)\) such that \(tu_{0}\in\mathcal{N}\), so that
where \(p_{i}\) is given by (2.22). Then (2.33) finishes the proof of Lemma 2.5. □
Lemma 2.6
There results \(\theta>0\).
Proof
If \(\theta=0\), then \(u=0\), which implies in particular that \(u_{n}\to 0\) in \(L^{p}_{\mathrm{loc}}(\mathbb{R}^{3})\) (\(1\le p<6\)). We first prove the following claim:
By (H2), for any \(\varepsilon>0\), there exists \(R_{\varepsilon}>0\) such that \(0<\alpha-V(x)\le\varepsilon, \forall|x|\ge R_{\varepsilon}\). Then
where
Noticing that
we have from (2.35) that
Since ε is arbitrarily small, (2.37) implies that (2.34), and then
where \(\epsilon_{n}\to0\) as \(n\to\infty\).
On the other hand, the fact \(u_{n}\in \mathcal{N}\) shows that
Denote
Since \(4< q< m\) and (2.39), it follows that \(\gamma_{n}(1)\ge0\) and \(\gamma_{n}(t)\to-\infty\) as \(t\to\infty\). Then there exists \(t_{n}\ge1\) such that \(\gamma_{n}(t_{n})=0\), and then \(t_{n}u_{n}\in \mathcal{N}_{\alpha}\), that is,
Using the facts that \(\|u_{n}\|_{Y}\) and \(\|u_{n}\|_{q,b}\) are bounded and \(\lim_{n\to\infty}\|u_{n}\|_{m,a}^{m}=\beta\), we deduce from (2.41) that the sequence \(\{t_{n}\}\) is bounded. If necessary, up to a subsequence, we can assume \(t_{n}\to t_{0}\ge1\). Then it follows from (2.38), (2.39) and (2.41) that
Since \(t_{n}^{4}\le t_{n}^{q}\) and \(F(u_{n})\ge0\), we have
Letting \(n\to\infty\) in (2.43) yields
where \(\nu=\limsup_{n\to\infty}\|u_{n}\|_{E}^{2}>0\) and \(\beta=\lim_{n\to\infty}\|u_{n}\|_{m,a}^{m}\). Since \(t_{0}\ge1\) and \(4< q< m\), we have \(t_{0}=1\). That is, \(t_{n}\to1\) as \(n\to\infty\). Therefore, it follows from (2.28) and (2.38) that
where \(\gamma_{n}=p_{1}\varepsilon_{n}t_{n}^{2}+p_{2}(t_{n}^{4}-t_{n}^{2})F(u_{n}) +p_{3}(t_{n}^{m}-t_{n}^{2})\|u_{n}\|_{m,a}^{m}\). Since \(t_{n}\to1\), we get \(\gamma_{n}\to0\). Moreover, the facts \(J(u_{n})\to d\) and \(t_{n}\to1\) in (2.45) imply that \(d_{\alpha}\le d\). This contradicts the result in Lemma 2.5. Therefore, we have \(\theta>0\) and complete the proof of Lemma 2.6. □
In the following, we consider the case \(\theta\in(0, \beta)\). As in [20, 21], we let \(\theta\in(0, \beta)\) and take \(\{u_{n}\}\) as a minimizing sequence for d on \(\mathcal{N}\), which satisfies (2.12) and
where and in the sequel, \(B_{r}=\{x\in\mathbb{R}^{3}:|x|< r\}\), \(B_{r}^{c}=\{x\in\mathbb{R}^{3}:|x|\ge r\}\), \(\Omega_{n}=\{x\in \mathbb{R}^{3}: r_{n}\le|x|< r_{n+1}\}\) with \(r_{n}\uparrow\infty\) and \(o_{n}(1)\) is a quantity which goes to zero as \(n\to\infty\).
Since \(J(u_{n})=J(|u_{n}|)\) in E, we assume \(u_{n}\ge0\) in \(\mathbb{R}^{3}\). Furthermore, we also consider, for every \(n\in\mathbb{N}\), a function \(\varphi_{n}\in C_{0}^{\infty}(\mathbb{R}^{3})\) such that
-
(1)
\(0\le\varphi_{n}(x)\le1\), \(\forall x\in \mathbb{R}^{3}\),
-
(2)
\(\varphi_{n}(x)=1\) if \(|x|\le r_{n}\), \(\varphi_{n}(x)=0\) if \(|x|\ge r_{n+1}\),
-
(3)
\(|\nabla\varphi_{n}(x)|\le C_{0}\), \(\forall x\in\mathbb {R}^{3}\) and \(\forall n\ge1\),
where \(C_{0}\) is some positive number independent of n. Furthermore, we set
Then \(u'_{n}, u''_{n}\ge0\) and \(u_{n}=u'_{n}+u''_{n}\) in \(\mathbb{R}^{3}\) for every \(n\ge1\).
Lemma 2.7
The following properties for \(u_{n}\), \(u'_{n}\), \(u''_{n}\) hold:
- (P1):
-
\(u'_{n}\rightharpoonup u\) weakly in \(E=W^{1,2}(\mathbb{R}^{3})\), \(u'_{n}\to u\) strongly in \(L^{m}(\mathbb{R}^{3},a)\cap L^{q}(\mathbb{R}^{3},b)\).
- (P2):
-
\(\int_{\mathbb{R}^{3}}a(x)|u_{n}|^{m}\,dx=\int_{\mathbb {R}^{3}}a(x)|u'_{n}|^{m}\,dx+\int_{\mathbb{R}^{3}}a(x)|u''_{n}|^{m}\,dx+o_{n}(1)\).
- (P3):
-
\(\int_{\mathbb{R}^{3}}b(x)|u_{n}|^{q}\,dx=\int_{\mathbb {R}^{3}}b(x)|u'_{n}|^{q}\,dx+\int_{\mathbb{R}^{3}}b(x)|u''_{n}|^{q}\,dx+o_{n}(1)\).
- (P4):
-
\(\|u_{n}\|_{E}^{2}\ge\|u'_{n}\|_{E}^{2}+\|u''_{n}\|_{E}^{2}+o_{n}(1)\).
- (P5):
-
\(F(u_{n})\ge F(u'_{n})+F(u''_{n})\), \(\forall n\ge1\).
- (P6):
-
\(F(u_{n})\le F(u'_{n})+F(u''_{n})+\sum_{j=1}^{5}A_{n}^{j}\), \(\forall n\ge 1\), in which
$$\begin{aligned}& A_{n}^{1}\equiv \int_{\Omega_{n}}K(x)\phi_{u_{n}}(x)u_{n}^{2}(x) \,dx=o_{n}(1), \\& A_{n}^{2}\equiv \int_{\Omega_{n}}K(x)\phi_{u'_{n}}(x)u_{n}^{2}(x) \,dx=o_{n}(1), \\& A_{n}^{3}\equiv \int_{\Omega_{n}}K(x)\phi_{u''_{n}}(x)u_{n}^{2}(x) \,dx=o_{n}(1), \\ & A_{n}^{4}\equiv \int_{B^{c}_{r_{n+1}}}K(x)\phi_{u'_{n}}(x)u_{n}^{2}(x) \,dx=o_{n}(1), \\ & A_{n}^{5}\equiv \int_{B_{r_{n}}}K(x)\phi_{u''_{n}}(x)u_{n}^{2}(x) \,dx=o_{n}(1),\quad n=1,2,\ldots, \end{aligned}$$(2.48)
where \(\phi_{u}\) is defined by (1.10).
Proof
The proof of properties (P1)-(P4) is similar to that in [20, 21] and is omitted. Here, we prove (P5) and (P6). Note that \(u'_{n}(x)=u_{n}(x)\), \(u''_{n}(x)=0\) if \(x\in B_{r_{n}}\) and \(u''_{n}(x)=u_{n}(x)\), \(u'_{n}(x)=0\) if \(x\in B_{r_{n+1}}^{c}\). So,
Moreover, if \(x\in\Omega_{n}\), we have \(0\le\varphi_{n}(x)\le1\) and
Thus, one sees that
Clearly,
So, we have
which proves (P5).
On the other hand, it follows from (2.52) that
Noticing that
we have from (2.54) and (2.55) that
If \(K\in L^{\infty}(\mathbb {R}^{3})\) and \(K\ge0\), we have from (2.12) that
From (2.46), we have \(\|u_{n}\|^{2}_{L^{\frac{12}{5}}(\Omega_{n})}=o_{n}(1)\) and so \(A_{n}^{1}=o_{n}(1)\).
If \(K\in L^{2}(\mathbb {R}^{3})\) and \(K\ge0\), we derive from the Hölder inequality and the Sobolev inequality that
Similarly, the assumption \(K\in L^{2}(\mathbb {R}^{3})\) implies that \(\|K\|_{L^{2}(\Omega_{n})}=o_{n}(1)\) and \(A_{n}^{1}=o_{n}(1)\).
Furthermore, since \(u_{n}(x)\ge u'_{n}(x), u''_{n}(x)\), we have \(\phi_{u_{n}}(x)\ge\phi_{u'_{n}}(x), \phi_{u''_{n}}(x)\), and then \(0\le A_{n}^{2}, A_{n}^{3}\le A_{n}^{1}=o_{n}(1)\). Noticing that
we argue as in the proof of (2.57) and (2.58) and obtain \(A_{n}^{4}=o_{n}(1)\). Similarly,
we get from (2.59) and \(A_{n}^{4}=o_{n}(1)\) that \(A_{n}^{5}=o_{n}(1)\) if \(K\in L^{2}(\mathbb {R}^{3})\cup L^{\infty}(\mathbb {R}^{3})\) and finish the proof of (P6). Then the proof of Lemma 2.7 is completed. □
3 Proof of Theorem 1.1
In order to prove Theorem 1.1, we first show that the weak limit u of the minimizing sequence \(\{u_{n}\}\) in Section 2 verifies
and
where \(p_{i}\) is given by (2.22).
Let \(\{u_{n}\}\) be a minimizing sequence for d on \(\mathcal{N}\) in Section 2. Then, if necessary, up to a subsequence, we have \(u_{n} \rightharpoonup u\) in E and \(L^{m}(\mathbb{R}^{3},a)\). By the weak lower semi-continuity of norms, we obtain
This implies relation (3.1). By Lemma 2.6, we get \(u\ge 0, u\not\equiv0\) in \(\mathbb{R}^{3}\). We now prove (3.2). If
we set
Then one sees that \(h(t)>0\) for small \(t>0\) and \(h(1)<0\), so that there exists \(t\in(0,1)\) such that \(h(t)=0\) and \(tu\in\mathcal{N}\) and
This contradiction shows that (3.4) cannot occur. Now we assume
and choose small \(\delta_{1}>0\) such that
Since \(u'_{n}\rightharpoonup u\) weakly in E, one has
Moreover, it follows from (P6) that
Then, from (3.8)-(3.10), there exist \(\delta_{2}\in (0,\delta_{1})\) and \(n_{0}\ge1\) such that
Now, by Lemma 2.7, we derive from (3.10), (3.11) and the fact \(u_{n}\in\mathcal{N}\) that
This implies that
The application of (2.38) and (3.13) shows that there exist \(\delta_{3}\in(0,\delta_{2})\) and \(n_{1}\ge n_{0}\) such that
Again, we set an auxiliary function
Obviously, \(\gamma_{n}(t)>0\) for small \(t>0\) and \(\gamma_{n}(1)<0\) for \(n\ge n_{1}\). Then there exists \(t_{n}\in(0,1)\) such that \(\gamma_{n}(t_{n})=0\) and \(t_{n}u''_{n}\in\mathcal{N}_{\alpha}\). Then it follows from (2.28) and (2.38) that
This shows that \(d_{\alpha}\le d\) and contradicts the result in Lemma 2.5. Therefore, (3.7) fails to be true and (3.2) is satisfied.
Since u satisfies (3.1) and (3.2), it follows that \(u\in\mathcal{N}\) and so that \(J(u)=d\) and u is a minimum point for J on \(\mathcal{N}\).
To finish the proof of Theorem 1.1, it is sufficient to prove that u is a critical point for the functional \(J(u)\) in E, that is, \(J'(u)v=0\) for all \(v\in E\), and thus \(J'(u)=0\) in \(E^{*}\).
For every fixed \(v\in E\), there exists \(\varepsilon>0\) such that \(u+sv \not\equiv0\) for all \(s\in(-\varepsilon,\varepsilon)\). From Lemma 2.1, there is \(t(s)\in\mathbb{R}\) such that \(t(s)(u+sv )\in \mathcal{N}\). We will show that \(t=t(s)\) is a \(C^{1}\) function. So, we consider the function \(\varphi: (-\varepsilon, \varepsilon)\times \mathbb{R}\rightarrow\mathbb{R}\) as
Since \(u\in\mathcal{N}\), we have
Moreover, we derive
Then, by the implicit function theorem, there exists a \(C^{1}\) function \(t(s)\) such that \(\varphi(s,t(s))=0\) and \(t(0)=1\) for every \(s\in(-\varepsilon_{0},\varepsilon_{0})\subset (-\varepsilon,\varepsilon)\). This also shows that \(t(s)\neq0\) and \(t(s)(u+sv)\in\mathcal{N}\). Then the function \(h(s)=J(t(s)(u+sv))\), \(s\in(-\varepsilon_{0}, \varepsilon_{0}) \) is in \(C^{1}\) and has a minimum at \(s=0\). Therefore,
Then, from (3.14), it follows for every \(v\in E\) that \(J'(u)v=0\). Thus \(J'(u)=0\) in \(E^{*}\). So, u is a critical point for J, and then u is a weak solution of problem (1.1) in E, that is, u is a solution of (1.1). Since \(J(u)=J(|u|)=d>0\), we can assume \(u\ge0\) a.e. in \(\mathbb{R}^{3}\). Furthermore, the application of maximum principle in [22] yields \(u(x)>0\) in \(\mathbb{R}^{3}\). Then the proof of Theorem 1.1 is finished.
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This work is supported by China Postdoctoral Science Foundation funded project (No. 2017M610436).
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Yuan, Q., Chen, C. & Yang, H. Existence of positive solutions for a Schrödinger-Poisson system with bounded potential and weighted functions in \(\mathbb{R}^{3}\) . Bound Value Probl 2017, 151 (2017). https://doi.org/10.1186/s13661-017-0886-6
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DOI: https://doi.org/10.1186/s13661-017-0886-6